It’s possible to come on a day trip from Kagoshima, but staying at least four days is ideal to enjoy what the island has to offer. It will give you more flexibility to plan around the weather too.

Itinerary
Day 1 Catch the ferry to Yakushima. Spend the afternoon exploring one of the villages, such as Shitoko and the Banyan Park.
Day 2 Hike Shiratani Unsuikyo or Yakusugi Land. Jomon-sugi is for committed hikers only, an 8 to 10 hour hike, catching the bus at 5 am.
Day 3 Sightseeing around the island and relax in an onsen (hot spring) If you have a car, you can drive all the way around the island. The bus runs most of the way around the island apart from the western, Seibu Rindoh Forest Path.
Day 4 Join a yakusugi craft workshop and make your own souvenir of the trip before your ferry back to Kagoshima.

Shiratani Unsuikyo
This is an enjoyable hike of three to four hours. To the moss forest and on a clear day, it’s worth walking the extra hour to Taikoiwa rock for a view from the top. If you are short of time, Yayoisugi Cedar Course is a shorter one hour hike.
Some of the paths or the park itself may close after heavy rain as the rivers become to dangerous to cross. So check in advance if it has been raining a lot.
How to get there:
It takes about 30 minutes by car from Miyanoura to the trail entrance. There is a direct bus from Miyanoura. Yakushima Kotsu bus leaves Miyanoura Port at 8:10, 10:15, 13:05, and 15:30. Matsubanda Kotsu bus leaves at 8:15, 12:25, and 13:40.

Yakusugi Land
Don’t be put off by the name. It’s actually an ideal place to see numerous giant old cedars along Arakawa River. There is a 30 minute course to three to four hour courses. A further 20 minute drive up from Yakusugi Land is Kigensugi Cedar, an ancient cedar that you can see up close.
How to get there:
There are two buses a day from Anbo to Yakusugi Land and Kigensugi, leaving Anbo at 9:20 and 13:30. It’s a 50 minute ride to Yakusugi Land. If you take the second bus, it gives you and hour to explore before the bus back at 15:15. Take the first bus to give you enough time to go see Kigensugi Cedar.

Jomonsugi Cedar
Standing at 25.3 meters in height, this supposedly the largest Yakusugi cedar in Japan. The photogenic tree is also the most popular destination for Japanese tourists. The path takes you through the site of Kosugidani village, where up to 540 people once lived, the men working in logging industry until 1970. Another iconic sight along the path is Wilson’s Stump.
How to get there:
Two buses in the morning, 4:00 and 4:45 from Miyanoura or 4:40 and 5:23 from Anbo takes you to Yakusugi Shizenkan Museum. From there, take the Arakawa Trekking Bus leaving at 5:00, 5:20, 5:40, and 6:00 (¥700, 30 minutes one-way)